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A Prudnikov sum is extended to derive the finite sum of the Hurwitz-Lerch Zeta function in terms of the Hurwitz-Lerch Zeta function. This formula is then used to evaluate a number trigonometric sums and products in terms of other trigonometric functions. These sums and products are taken over positive integers which can be simplified in certain circumstances. The results obtained include generalizations of linear combinations of the Hurwitz-Lerch Zeta functions and involving powers of 2 evaluated in terms of sums of Hurwitz-Lerch Zeta functions. Some of these derivations are in the form of a new recurrence identity and finite products of trigonometric functions.

In this paper, we derive an integral transform involving the product of Hermite polynomial $ H_{n}(\beta x) $ and parabolic cylinder function $ D_{v}(\alpha t) $. These integral transforms will be evaluated in terms of Lerch function. Various formulae are also evaluated in terms of special functions to complete this paper. All the results in this paper are new.

In this paper we use a contour integral method to derive a bilateral generating function in the form of a double series involving Chebyshev polynomials expressed in terms of the incomplete gamma function. Generating functions for the Chebyshev polynomial are also derived and summarized. Special cases are evaluated in terms of composite forms of both Chebyshev polynomials and the incomplete gamma function.

We evaluate definite integrals of the form given by $\int_{0}^{\infty}R(a, x)\log (\cos (\alpha) \text{sech}(x)+1)dx$. The function $R(a, x)$ is a rational function with general complex number parameters. Definite integrals of this form yield closed forms for famous integrals in the books of Bierens de Haan [4] and Gradshteyn and Ryzhik [5].

In this work we present derivations of the formula listed in entry 4.113 in the sixth edition of Gradshteyn and Rhyzik's table of integrals. We evaluate two definite integrals of the formandin terms of the Lerch function where $ k $, $ a $, $ z $ and $ b $ are arbitrary complex numbers. The entries in the table(s) are obtained as special cases in the paper below.

In 1974 Fritz Oberhettinger published the Tables of Mellin Transforms. We derive the correct result for one of these integrals along with some interesting special cases.

With a possible connection to integrals used in General Relativity, we used our contour integral method to write a closed form solution for a quadruple integral involving exponential functions and logarithm of quotient radicals. Almost all Hurwitz–Lerch Zeta functions have an asymmetrical zero distribution. All the results in this work are new.

Closed expressions for the integralare given where the variables $ a $, $ k $, $ m $ and $ u $ are general complex numbers. Some of these closed expressions are given in [4]. Some special cases of the integral are derived and discussed.

In this paper by means of contour integration we will evaluate definite integrals of the form \begin{equation*} \int_{0}^{1}\left(\ln^k(ay)-\ln^k\left(\frac{a}{y}\right)\right)R(y)dy \end{equation*} in terms of a special function, where $R(y)$ is a general function and $k$ and $a$ are arbitrary complex numbers. These evaluations can be expressed in terms of famous mathematical constants such as the Euler's constant $\gamma$ and Catalan's constant $C$. Using derivatives, we will also derive new integral representations for some Polygamma functions such as the Digamma and Trigamma functions along with others.

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form $\int_{0}^{\infty}\ln^k(\alpha y)\ln(R(y))dy$ in terms of a special function, where $R(y)$ is a general function and $k$ and $\alpha$ are arbitrary complex numbers.

The Wang sum involving the exponential sums of Lerch's Zeta functions is extended to the finite sum of the Huwitz-Lerch Zeta function to derive sums and products involving cosine and tangent trigonometric functions. The general theorem used to derive these sums and products is in the form of the finite sum over positive integers of the Hurwitz-Lerch Zeta function where the associated parameters are general complex numbers. New Hurwitz-Lerch Zeta function recurrence identities with consecutive neighbours are derived. Some finite sum and product formulae examples involving cosine, tangent and the product of cosine and tangent functions are also derived and evaluated.

We consider a Hurwitz-Lerch zeta function Φs,z,a sum over the natural numbers. We provide an analytically continued closed form solution for this sum in terms of the addition of Hurwitz-Lerch zeta functions. A new recurrence identity with consecutive neighbours and the product of trigonometric functions is derived.

A two-dimensional integral containing Sx is derived. Sx is the Fresnel integral function, and the double integral is taken over the range 0

Using a contour integral method, the de Montmort-Prudnikov sum is extended to derive a new series representation involving the incomplete gamma function. The series is uniformly convergent and completely analytical, which can be evaluated for general complex ranges of the parameters involved. Applications and evaluations of this formula are discussed.

This paper employs a contour integral method to derive and evaluate the infinite sum of the Euler polynomial expressed in terms of the Hurwitz Zeta function. We provide formulae for several classes of infinite sums of the Euler polynomial in terms of the Riemann Zeta function and fundamental mathematical constants, including Catalan’s constant. This representation of Catalan’s constant suggests it could be rational.

A family of generalized definite logarithmic integrals given by $$ \int_{0}^{1}\frac{\left(x^{ i m} (\log (a)+i \log (x))^k+x^{-i m} (\log (a)-i \log (x))^k\right)}{(x+1)^2}dx$$ built over the Lerch function has its analytic properties and special values listed in explicit detail. We use the general method as given in [6] to derive this integral. We then give a number of examples that can be derived from the general integral in terms of well known functions.

A new three-dimensional integral containing f(x,y,z)Iv(xα) is derived where Iv(xα) is the Modified Bessel Function of the first kind and the integral is taken over the infinite cubic space 0

We derive a closed-form expression for the infinite sum of the Hurwitz–Lerch zeta function using contour integration. This expression is used to evaluate infinite sum and infinite product formulae involving trigonometric functions expressed in terms of fundamental constants. These types of infinite sums and products have previously been and are currently studied by many mathematicians including Leonhard Euler. The results presented in this paper extend previous work by squaring parameters in the infinite sum of the Hurwitz–Lerch zeta function. This formula allows for new derivations featuring trigonometric functions with angles of powers of 2. The zero distribution of almost all Hurwitz–Lerch zeta functions is asymmetrical. A table of infinite products is produced highlighting the usefulness of this work and for easy reading by researchers interested in such formulae. Mathematica software was used in assisting with the numerical verification of the results in the tables produced.

The Ohtsuka–Vălean sum is extended to evaluate an extensive number of trigonometric and hyperbolic sums and products. The sums are taken over finite and infinite domains defined in terms of the Hurwitz–Lerch zeta function, which can be simplified to composite functions in special cases of integer values of the parameters involved. The results obtained include generalizations of finite and infinite products and sums of tangent, cotangent, hyperbolic tangent and hyperbolic cotangent functions, in certain cases raised to a complex number power.

In this manuscript, the authors derive closed formula for definite integrals of combinations of powers and logarithmic functions of complicated arguments and express these integrals in terms of the Hurwitz zeta functions. These derivations are then expressed in terms of fundamental constants, elementary, and special functions. A summary of the results is produced in the form of a table of definite integrals for easy referencing by readers.

Upon reading the famous book on integral transforms volume II by Erdeyli et al., we encounter a formula which we use to derive a Mellin transform given by ∫0∞xm−1logkax/β2+x2γ+xdx, where the parameters a,k,β, and γ are general complex numbers. This Mellin transform will be derived in terms of the Lerch function and is not listed in current literature to the best of our knowledge. We will use this transform to create a table of definite integrals which can be used to extend similar tables in current books featuring such formulae.

Closed expressions for a number of septuple integrals involving the product of three Bessel functions of the first kind Jα(tβ)Jγ(xδ)Jη(yθ) when the orders α,γ,η are large, are derived in terms of the Hurwitz–Lerch zeta function Φ(z,s,v). The integrals are not easy to to evaluate for complex values of the parameters. All the results in this work are new.

In the fields of science and engineering, tasks involving repeated integrals appear on occasion. The authors’ study on repeated integrals of a class of exponential and logarithmic functions is presented in this publication. The paper includes several examples that demonstrate the evaluation of the analytical parts of the multi-dimensional integral derived. All the results in this work are new.

A four-dimensional integral containing g(x,y,z,t)Cn(λ)(x) is derived. Cn(λ)(x) is the Gegenbauer polynomial, g(x,y,z,t) is a product of the generalized logarithm quotient functions and the integral is taken over the region 0≤x≤1,0≤y≤1,0≤z≤1,0≤t≤1. The integral is difficult to compute in general. Special cases are given and invariant index forms are derived. The zero distribution of almost all Hurwitz–Lerch zeta functions is asymmetrical. All the results in this work are new.

The aim of the current document is to evaluate a quadruple integral involving the Chebyshev polynomial of the first kind Tn(x) and derive in terms of the Hurwitz-Lerch zeta function. Special cases are evaluated in terms of fundamental constants. The zero distribution of almost all Hurwitz-Lerch zeta functions is asymmetrical. All the results in this work are new.

We examine the improved infinite sum of the incomplete gamma function for large values of the parameters involved. We also evaluate the infinite sum and equivalent Hurwitz-Lerch zeta function at special values and produce a table of results for easy reading. Almost all Hurwitz-Lerch zeta functions have an asymmetrical zero distribution.

The objective of the present paper is to obtain a quadruple infinite integral. This integral involves the product of the Struve and parabolic cylinder functions and expresses it in terms of the Hurwitz–Lerch Zeta function. Almost all Hurwitz-Lerch Zeta functions have an asymmetrical zero distributionSpecial cases in terms fundamental constants and other special functions are produced. All the results in the work are new.

In this paper, we have derived and evaluated a quadruple integral whose kernel involves the logarithm and product of Bessel functions of the first kind. A new quadruple integral representation of Catalan’s G and Apéry’s ζ(3) constants are produced. Some special cases of the result in terms of fundamental constants are evaluated. All the results in this work are new.

A closed form expression for a triple integral not previously considered is derived, in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. The kernel of the integral involves the product of the logarithmic, exponential, quotient radical, and polynomial functions. Special cases are derived in terms of fundamental constants; results are summarized in a table. All results in this work are new.

We derive a new formula for the Hurwitz–Lerch zeta function in terms of the infinite sum of the incomplete gamma function. Special cases are derived in terms of fundamental constants.

We present a method using contour integration to derive definite integrals and their associated infinite sums which can be expressed as a special function. We give a proof of the basic equation and some examples of the method. The advantage of using special functions is their analytic continuation, which widens the range of the parameters of the definite integral over which the formula is valid. We give as examples definite integrals of logarithmic functions times a trigonometric function. In various cases these generalizations evaluate to known mathematical constants, such as Catalan’s constant C and π.

In this work, the authors use their contour integral method to derive an application of the Fourier integral theorem given by ∫−∞∞∫−∞∞emx−my−ex−ey+y(log(a)+x−y)kdxdy in terms of the Lerch function. This integral formula is then used to derive closed solutions in terms of fundamental constants and special functions. Almost all Lerch functions have an asymmetrical zero distribution. There are some useful results relating double integrals of certain kinds of functions to ordinary integrals for which we know no general reference. Thus, a table of integral pairs is given for interested readers. All of the results in this work are new.

The object of this paper is to derive a double integral in terms of the Hurwitz–Lerch zeta function. Almost all Hurwitz–Lerch zeta functions have an asymmetrical zero-distribution. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.

A class of definite integrals involving a quotient function with a reducible polynomial, logarithm and nested logarithm functions are derived with a possible connection to contact problems for a wedge. The derivations are expressed in terms of the Lerch function. Special cases are also derived in terms fundamental constants. The majority of the results in this work are new.

A quadruple integral involving the logarithmic, exponential and polynomial functions is derived in terms of the Lerch function. Special cases of this integral are evaluated in terms of special functions and fundamental constants. Almost all Lerch functions have an asymmetrical zero-distribution. The majority of the results in this work are new.

In this manuscript, the authors derive a double integral whose kernel involves the logarithmic function a polynomial raised to a power and a quotient function expressed it in terms of the Lerch function. All the results in this work are new.

This is a compilation of definite integrals of the product of the hyperbolic cosecant function and polynomial raised to a general power. In this work, we used our contour integral method to derive a Fourier sine transform in terms of the Lerch function. Almost all Lerch functions have an asymmetrical zero-distribution. A summary table of the results are produced for easy reading. A vast majority of the results are new.

We apply our simultaneous contour integral method to an infinite sum in Prudnikov et al. and use it to derive the infinite sum of the Incomplete gamma function in terms of the Hurwitz zeta function. We then evaluate this formula to derive new series in terms of special functions and fundamental constants. All the results in this work are new.

This paper gives new integrals related to a class of special functions. This paper also showcases the derivation of definite integrals involving the quotient of functions with powers and the exponential function expressed in terms of the Lerch function and special cases involving fundamental constants. The goal of this paper is to expand upon current tables of definite integrals with the aim of assisting researchers in need of new integral formulae.

This is a collection of definite integrals involving the logarithmic and polynomial functions in terms of special functions and fundamental constants. All the results in this work are new.

In this present work we derive, evaluate and produce a table of definite integrals involving logarithmic and exponential functions. Some of the closed form solutions derived are expressed in terms of elementary or transcendental functions. A substantial part of this work is new.

In this article we derive some entries and errata for the book of Gradshteyn and Ryzhik which were originally published by Bierens de Haan. We summarize our results using tables for easy reading and referencing.

This manuscript concerns two definite integrals that could be connected to the Bose-Einstein and the Fermi-Dirac functions in the integrands, separately, with numerators slightly modified with a difference in two expressions that contain the Fourier kernel multiplied by a polynomial and its complex conjugate. In this work, we use our contour integral method to derive these definite integrals, which are given by ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx−1dx and ∫0∞ie−imx(log(a)−ix)k−eimx(log(a)+ix)k2eαx+1dx in terms of the Lerch function. We use these two definite integrals to derive formulae by Erdéyli and Watson. We derive special cases of these integrals in terms of special functions not found in current literature. Special functions have the property of analytic continuation, which widens the range of computation of the variables involved.

While browsing through the famous book of Bierens de Haan, we came across a table with some very interesting integrals. These integrals also appeared in the book of Gradshteyn and Ryzhik. Derivation of these integrals are not listed in the current literature to best of our knowledge. The derivation of such integrals in the book of Gradshteyn and Ryzhik in terms of closed form solutions is pertinent. We evaluate several of these definite integrals of the form ∫0∞(a+y)k−(a−y)keby−1dy, ∫0∞(a+y)k−(a−y)keby+1dy, ∫0∞(a+y)k−(a−y)ksinh(by)dy and ∫0∞(a+y)k+(a−y)kcosh(by)dy in terms of a special function where k, a and b are arbitrary complex numbers.

The derivation of integrals in the table of Gradshteyn and Ryzhik in terms of closed form solutions is always of interest. We evaluate several of these definite integrals of the form ∫ 0 ∞ log ( 1 ± e − α y ) R ( k , a , y ) d y in terms of a special function, where R ( k , a , y ) is a general function and k, a and α are arbitrary complex numbers, where R e ( α ) > 0 .